Find the zeros of a quadratic function from its graph'

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Objectives
Find the zeros of a quadratic function from its
graph. Find the axis of symmetry and the vertex
of a parabola.
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Vocabulary
zero of a function axis of symmetry
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Recall that an x-intercept of a function is a
value of x when y 0. A zero of a function is an
x-value that makes the function equal to 0. So a
zero of a function is the same as an x-intercept
of a function. Since a graph intersects the
x-axis at the point or points containing an
x-intercept, these intersections are also at the
zeros of the function. A quadratic function may
have one, two, or no zeros.
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Example 1A Finding Zeros of Quadratic Functions
From Graphs
Find the zeros of the quadratic function from its
graph. Check your answer.
y x2 2x 3
The zeros appear to be 1 and 3.
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Example 1C Finding Zeros of Quadratic Functions
From Graphs
Find the zeros of the quadratic function from its
graph. Check your answer.
y 2x2 2
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A vertical line that divides a parabola into two
symmetrical halves is the axis of symmetry. The
axis of symmetry always passes through the vertex
of the parabola. You can use the zeros to find
the axis of symmetry.
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Example 2 Finding the Axis of Symmetry by Using
Zeros
Find the axis of symmetry of each parabola.
A.
(1, 0)
Identify the x-coordinate of the vertex.
The axis of symmetry is x 1.
B.
Find the average of the zeros.
The axis of symmetry is x 2.5.
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Check It Out! Example 2
Find the axis of symmetry of each parabola.
a.
(3, 0)
Identify the x-coordinate of the vertex.
The axis of symmetry is x 3.
b.
Find the average of the zeros.
The axis of symmetry is x 1.
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If a function has no zeros or they are difficult
to identify from a graph, you can use a formula
to find the axis of symmetry. The formula works
for all quadratic functions.
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Example 3 Finding the Axis of Symmetry by Using
the Formula
Find the axis of symmetry of the graph of y
3x2 10x 9.
Step 1. Find the values of a and b.
y 3x2 10x 9
a 3, b 10
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Check It Out! Example 3
Find the axis of symmetry of the graph of y
2x2 x 3.
Step 1. Find the values of a and b.
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Once you have found the axis of symmetry, you can
use it to identify the vertex.

• Find the axis of symmetry.
• Substitute x-value into function to find y-value.
• Write as an ordered pair.

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Example 4A Finding the Vertex of a Parabola
Find the vertex.
y 0.25x2 2x 3
Step 1 Find the x-coordinate of the vertex. The
zeros are 6 and 2.
Step 2 Find the corresponding y-coordinate.
Use the function rule.
y 0.25x2 2x 3
0.25(4)2 2(4) 3 1
Substitute 4 for x .
The vertex is (4, 1).
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Example 4B Finding the Vertex of a Parabola
Find the vertex.
y 3x2 6x 7
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Check It Out! Example 4
Find the vertex.
y x2 4x 10
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Assignment

• p. 603 19-35 odd
• 47-51 odd